ON A METHOD FOR FINDING THE STEADY-STATE SOLUTION FOR SELF-OSCILLATORS

M. S. Roitman

(Presented by the scientific seminar of the Department of Radio Engineering)

A self-oscillator* can, by means of structural transformations, be reduced to the structural diagram shown in Fig. 1. It is described by a nonlinear differential equation of order \(m\) \((m \geqslant 2)\)

\[ W[E(x,\dot{x}\ldots)]-1=0. \tag{1} \]

For an approximate solution of a second-order system with one nonlinear element, the well-known methods of slowly varying amplitudes (developed by Van der Pol [1], L. I. Mandelstam, N. D. Papaleksi [2]) and of the small parameter** (proposed by A. A. Andronov on the basis of the work of H. Poincaré [3]) are used.

However, it should be noted that the method of slowly varying amplitudes is applicable only when one may neglect both nonlinear distortions of the waveform and nonlinear frequency shifts.

Fig. 1.

A number of modifications of the small-parameter method are set forth in [4]. For the investigation of nonlinear systems of the second order, the method of variable scale can also be used effectively (it is given with great completeness in the monograph [5]).

The phase-space method is very illustrative (developed by H. Leautaud [6], A. Liénard [7], A. A. Andronov, A. A. Vitt, and S. E. Khaikin [3]). Various modifications of harmonic linearization of nonlinear systems*** [8, 9, 10, 11, etc.] are finding ever wider application, especially in automatic control (its foundations were laid in the works of N. M. Krylov and N. N. Bogolyubov). The advantage of harmonic linearization is, above all, that it makes possible the analysis of systems of any order with one nonlinear element. However, obtaining refined solutions is associated with cumbersome computations (for example, the methods of Yu. N. Topcheev) and with little visual clarity.

* In the present work, self-oscillators based on amplifying links are considered, and not two-terminal networks with negative resistance.

** The small-parameter method is, in principle, also applicable to the analysis of systems above the second order.

*** The analysis of a self-oscillator by the method of equivalent linearization of tube transconductance was first carried out by Yu. B. Kobzarev [12].

Let us note that, in analyzing automatic-control systems, one is interested in a refined solution primarily for the fundamental harmonic. We, however, are also very interested in the nonlinear distortions at the generator output.

Therefore, despite the relatively well-developed theory of quasi-linear self-oscillatory systems, the author considered it justified to attempt to simplify the analysis and make it more transparent.

The structure of a self-oscillator can be reduced to the form shown in Fig. 2, where \(W_{l1}, W_{l2}\) are linear elements, and \(W_{\mathrm{n}}\) is a nonlinear frequency-independent element.

Fig. 2.

When a sinusoidal signal \(U_m \sin \omega t\) is applied to the input of the nonlinear element (point \(a\)), the first harmonic of the output voltage is equal to

\[ U_{m\text{out}}\sin \omega t = \frac{1}{2\pi}\cdot \int_{0}^{2\pi} W_{\mathrm{n}}[U_m\sin \omega t]\,U_m\sin \omega t\,d\omega t . \]

The ratio

\[ \frac{U_{m\text{out}}}{U_m}=W_{\mathrm{n}}(u) \tag{2} \]

will be called the linearized (averaged) transfer coefficient of the nonlinear element.

The quantity \(W_{\mathrm{n}}(u)\) now depends on the amplitude (mean, effective) value of the voltage, but not on the instantaneous value.

Equation (1) can be rewritten in the form

\[ W_{l1}(\omega)e^{j\varphi_1(\omega)} \cdot W_{\mathrm{n}}(u)\cdot W_{l2}(\omega)e^{j\varphi_2(\omega)} -1=0 . \]

Changing the notation to that more customary for a radio engineer,

\[ \dot W_{l1}(\omega)=\dot\beta_n(\omega) \quad\text{and}\quad W_{\mathrm{n}}(u)\cdot \dot W_{l2}(\omega) = K_{\mathrm{n}}(u)\cdot \dot K_{\mathrm{y}}(\omega) = \dot K(u)(\omega), \]

we obtain

\[ K(u)(\omega)\cdot \beta_n(\omega) e^{j[\varphi_k(\omega)+\varphi_\beta(\omega)]} -1=0 . \]

The system will generate at frequencies \(\omega\) for which the well-known phase-balance conditions are satisfied,

\[ \varphi_k(\omega)+\varphi_\beta(\omega) = \varphi_\Sigma(\omega) = 2n\pi \quad (n=0,1,2\ldots) \tag{3} \]

and the amplitudes

\[ K(u)(\omega)\cdot \beta(\omega)>1 . \tag{4} \]

If conditions (3), (4) are satisfied only at one frequency, then the oscillations will be quasi-harmonic. The generation frequency \((\omega_0)\) is determined, in the first approximation, from (3), and the oscillation amplitude from the equation

\[ K(u)(\omega)\cdot \beta(\omega)=1 . \tag{5} \]

It is easy to see that, for stability of the stationary solution, the condition

\[ \frac{dK(u)}{du}<0 \tag{6} \]

must be satisfied.

Formulas (2, 3, 5) are valid regardless of where the nonlinear element is located (provided, of course, that the nonlinearity of the input and output resistances of this element has little effect on the system, as is usually attempted in practice).

The equivalent quality factor of the system is determined from the expression

\[ Q_{\mathrm{e}}=\frac{d\varphi_\Sigma(\omega)}{d\omega}\cdot\frac{\omega}{2}. \]

After finding the amplitude and frequency of oscillation, we determine, in the first approximation, the coefficients of the generator harmonics, regarding it as an amplifier with positive feedback [13]

\[ \dot{k}_{n\mathrm{g}}=\frac{\dot{k}_{ny}}{1-\dot{\beta}_n(n\omega)\cdot \dot{K}(u)(n\omega)}, \tag{7} \]

where

\[ \dot{k}_{ny} \]

is the coefficient of the \(n\)-th harmonic of the amplifier without positive feedback;

\[ \dot{\beta}_n(n\omega),\ \dot{K}(u)(n\omega) \]

are the transfer coefficients, for the \(n\)-th harmonic, of the positive-feedback circuit and of the amplifier, respectively.

Such a transition not only simplifies the determination of distortion, but also sharply simplifies the finding, when necessary, of a refined solution.

In practice, in calculations one may substitute \(K(n\omega)\) in (7) instead of \(K(u)(n\omega)\) (i.e., consider the amplifier linear).

Thus, the procedure for finding the first approximation is as follows:

1) determine the oscillation frequency from (3);

2) find the linearized value of the transfer coefficient of the nonlinear element;

3) from equality (5) determine the amplitude of the input signal \(U_m\), and then also the amplitude of the first harmonic \(U_{m\mathrm{out}}\) at the generator output

\[ K(u)\cdot U_m=U_{m\mathrm{out}}; \]

4) compute the nonlinear distortions of the generator by formula (7).

In most cases the first approximation is quite sufficient, and striving to increase the accuracy of the calculations is not justified. But sometimes finding a refined solution is fundamentally necessary. As applied to amplitude-stable generators, this must be done when the nonlinearity is of an odd character. In view of the importance of the question raised, let us consider it in more detail.

The function of a nonlinear element \(f(x)\) (a tube, transistor, diode) may be represented in the form of a Taylor or Maclaurin series*

\[ f(x)=f(0)+\frac{x}{1!}\cdot\frac{df(0)}{dx}+\frac{x^2d^2f(0)}{2!\,dx^2}+\cdots+\frac{x^i}{i!}\frac{d^if(0)}{dx^i} \]

with remainder

\[ R_n=\frac{x^{i+1}}{(i+1)!}\cdot\frac{d^{i+1}f(\Theta x)}{dx^{i+1}}\quad (0<\Theta<1), \]

i.e., by a polynomial of \(n\)-th order

\[ f(x)=K_0+K'x+K''x^2+\cdots+K^ix^i, \tag{8} \]

where

\[ K=\frac{df(0)}{dx};\quad K''=\frac{1}{2}\cdot\frac{d^2f(0)}{dx^2};\quad K^i=\frac{1}{i!}\cdot\frac{d^if(0)}{dx^i}. \]

For electronic elements, series (8) is rapidly convergent, and in the analysis of self-oscillators it is permissible to restrict oneself to the first

* Since we are interested in wide-band generators, the nonlinearities may be regarded as weakly dependent on frequency and, consequently, as functions of only one variable.

by third-order terms*, which give a complete qualitative picture of the NE (constant, even, and odd components).

The error allowed in this case is usually smaller than the existing spread of parameters.

Thus, approximating the transfer coefficient of the nonlinear element \(K_{\mathrm N}\) in the form of the series (8) and applying \(U_m \sin \omega t\) to its input, we obtain at the output:

\[ \begin{aligned} u_{\text{out}}(t) &= U_m \sin \omega t \cdot K_{\mathrm N}[U_m \sin \omega t] \\ &= \left(K_{\mathrm N}+\frac{3}{4}K_{\mathrm N}''U_m^2\right)\cdot U_m \sin \omega t +\frac{1}{2}K_{\mathrm N}'U_m^2 -\frac{1}{2}K_{\mathrm N}'U_m^2\cos 2\omega t -\frac{1}{4}K_{\mathrm N}''\cdot \end{aligned} \tag{9} \]

\[ \cdot U_m^3 \sin \omega t = \sum_{n=1}^{3} A_{n\mathrm N}\sin(n\omega t+\varphi_n). \]

The linearized value of the amplification coefficient is equal to

\[ K(u)(\omega)= \left[ K_{\mathrm N}+\frac{3}{4}K_{\mathrm N}''U_m^2 \right]K_{\ell 2}(\omega). \tag{10} \]

It is easy to see that

\[ \frac{dK(u)}{du}=0, \]

if the even derivatives are absent in the expansion (8), i.e.,

\[ \frac{d^2 f(0)}{dx^2}=0,\qquad \frac{d^4 f(0)}{dx^4}=0,\ldots \]

But if

\[ \frac{dK(u)}{du}=0, \]

then stable generation is impossible (for the time being we are speaking of systems with inertialess nonlinearities (INE)).

In order to make a final judgment about the stability of the periodic regime, it is necessary to find its refined solution.

Thus, the following rule may be formulated (it will be refined): if, in the open-loop system of a self-oscillator, when a sinusoidal signal is applied to its input, odd harmonics** are absent at the output, then it is mandatory to determine the refined solution.

To find the nonlinear distortions and the refined solution, we shall use the following device.

Let us apply to the input of the amplifier a sinusoidal voltage from a source with internal resistance equal to the output section \(\beta_{\mathrm p}(\omega)\). The transfer coefficient for the fundamental harmonic is determined from (10), while the coefficients of the higher harmonics of the amplifier are equal to

\[ \dot K_{n\mathrm y} = \frac{\dot K_{\mathrm n2}(n\omega)}{\dot K_{\mathrm n2}(\omega)} \cdot K_{n(\mathrm{NE})} \approx \frac{\dot K(u)(n\omega)}{\dot K(u)(\omega)} \cdot K_{n(\mathrm{NE})}, \tag{11} \]

where

\[ K_{n(\mathrm{NE})} = \frac{1}{U_m\cdot K_{\mathrm N}\pi} \int_{0}^{2\pi} K_{\mathrm N}[U_m\sin\omega t]\, U_m\sin n\omega t\,d\omega t. \]

* Taking higher nonlinearities into account is strictly necessary in all those cases when part of the interference signal (or interferences), due to combination terms, can fall into the transparency band of the device, and this fact is essential (for example, in receivers of radio-reconnaissance equipment). In some cases, for example when reference diodes are used, it is more appropriate to employ a piecewise-linear approximation. But whatever the approximation of the NE may be, the conclusions drawn below remain valid.

** Directly from (9) it is seen that

\[ \frac{dK(U)}{dU}\ne 0, \]

if odd harmonics are present, i.e., if

\[ \frac{d^2 f(0)}{dx^2}\ne 0. \]

The harmonic coefficients for the generator are obtained by substituting the value \(\dot K_{ny}\) into expression (7). At the generator output we shall have

\[ u_{\text{out}}(t)=\sum_{n=1}^{q} A_n \sin(n\omega t+\varphi_n). \]

A refined solution can be found by analyzing the combination products. The voltages of the harmonics from the generator output, through the positive-feedback circuit 1, again arrive at the input of the nonlinear element. We must now find the complex transfer coefficient of the NE, taking into account that, instead of the sinusoidal signal \(U_m \sin \omega t\), it is necessary to substitute into \(W_{\text{n}}[u(t)]\) the spectrum

\[ u(t)=\sum_{n=1}^{q} A_n \beta_n(n\omega)\sin[n\omega t+\varphi_n+\varphi_\beta(n\omega)]. \tag{12} \]

The transfer coefficient of the NE is

\[ \dot K_{\text{n, ref}}(u)=K_{\text{n, ref}}(u)e^{j\varphi_{\text{ne}}} =\frac{U_{m\text{ out}}}{U_{tm}}e^{j\varphi_{\text{ne}}} =K_{\text{n}}(u)+\Delta_1K_{\text{n}}(u)+j\Delta_2K_{\text{n}}(u), \tag{13} \]

where \(\Delta_1K(u)\), \(\Delta_2K(u)\) are additional components of the transfer coefficient, caused by the appearance in the output-voltage spectrum of combination components of the fundamental frequency.

All the calculations for determining \(\Delta_1K_{\text{n}}(u)\) and \(\Delta_2K_{\text{n}}(u)\) can be sharply simplified if one uses the rules of combinatorics and takes into account that

\[ \beta[(n-1)\omega]>\beta(n\omega)_{n\ge 2}. \tag{14} \]

The additional component of the first harmonic will be produced by combinations of terms of the series (8) and (12) whose degrees are equal to

\[ |n-i|_{\substack{i\ne 0\\ n\ne 0}}=1. \]

But, owing to the attenuation of the nonlinearity* and condition (14), the principal contribution is made by the component \((n-i)=(2-1)\).

The basis for a further simplification of the analysis is the really existing limitation for all the generators under consideration—the smallness of the nonlinear distortions at the output.

Consequently, the “nonlinear mixing” of the phase of the first harmonic of the voltage at the NE output relative to the output is small, and with sufficient justification one may assume

\[ \cos\varphi_{\text{ne}}\simeq 1-\frac12\varphi_{\text{ne}}^2\simeq 1. \]

Then

\[ K_{\text{n, ref}}(u)\simeq K_{\text{n}}(u)+\Delta_1K_{\text{n}}(u), \]

and

\[ \varphi_{\text{ne}}=\arcsin \frac{\Delta_2K_{\text{n}}(u)} {K_{\text{n}}(u)+\Delta_1K_{\text{n}}(u)} \simeq \arcsin \frac{\Delta_2K(u)^{**}}{K_{\text{n}}(u)}. \]

Taking into account that the total phase shift for the first harmonic is equal to zero [see (3)] and

\[ \dot U_m=\dot U_m \dot K_{\text{n}}(u)\dot k_{\ell 2}(\omega)\dot\beta(\omega)=A_1\dot\beta(\omega), \]

\[ \text{* For definiteness, we stipulate that by attenuation of the nonlinearity we mean the conditions } \frac{d}{dU}K'_{\text{n}}U_m> \frac{d}{dU}K'''_{\text{n}}U_m^3 \text{ and } \frac{d}{dU}K''_{\text{n}}U_m^2> \frac{d}{dU}K''''_{\text{n}}U_m^4. \]

\[ \text{** The simplifications adopted are not fundamentally necessary.} \]

the correction to the amplitude and frequency of the fundamental harmonic of the oscillations is determined from the equalities

\[ \frac{\Delta_1 K_{\mathrm n}(u)}{K_{\mathrm n}(u)} \approx \frac{1}{U_m K_{\mathrm n}(u)\pi} \int_0^{2\pi} K_{\mathrm n}'\left[ U_m\sin\omega t - \operatorname{Re} \frac{A_{2\mathrm n}\dot K_{\mathrm l2}(2\omega)} {1-\dot K(u)(2\omega)\dot\beta(2\omega)} \cdot \dot\beta(2\omega)\cos 2\omega t \right]^2 \sin\omega t\,d\omega t = \frac{K_{\mathrm n}'}{K_{\mathrm n}(u)} \cdot A_{2\mathrm n}\operatorname{Re} \frac{\dot K(u)(2\omega)\dot\beta(2\omega)} {K_{\mathrm n}(u)[1-\dot K(u)(2\omega)\dot\beta(2\omega)]}; \tag{15} \]

\[ \frac{\Delta_2 K(u)}{K_{\mathrm n}(u)} = \frac{1}{U_m K_{\mathrm n}(u)\pi} \int_0^{2\pi} K_{\mathrm n}'\left[ U_m\sin\omega t+\operatorname{Im} \cdot \frac{A_{2\mathrm n}\dot K_{\mathrm l2}(2\omega)} {1-\dot K(u)(2\omega)\dot\beta(2\omega)} \cdot \dot\beta(2\omega)\sin 2\omega t \right]^2 \cos\omega t\,d\omega t = \]

\[ = \frac{K_{\mathrm n}'}{K_{\mathrm n}(u)} A_{2\mathrm n}\operatorname{Im} \frac{\dot K(u)(2\omega)\dot\beta(2\omega)} {K_{\mathrm n}(U)[1-\dot K(u)(2\omega)\dot\beta(2\omega)]}. \tag{16} \]

But

\[ A_{2\mathrm n} = \frac12 |K_{\mathrm n}'|U_m^2 = \frac12\left|\frac{dK_{\mathrm n}[f(u)]}{du}\right|U_m^2 = k_{2\mathrm n}A_{1\mathrm n} = k_{2\mathrm n}K_{\mathrm n}(u)U_m, \]

where \(k_{2\mathrm n}\) is the coefficient of the second harmonic at the output of the nonlinear element;

\(A_{1\mathrm n}\) is the amplitude of the first harmonic.

\[ K_{\mathrm n}' = \frac{2k_{2\mathrm n}K(u)}{U_m}\cdot \operatorname{sign} \frac{dK[f(u)]}{du} \]

and formulas (15), (16) can be written in the form

\[ \frac{\Delta_1 K_{\mathrm n}(u)}{K_{\mathrm n}(u)} = \frac12\cdot \frac{1}{K_{\mathrm n}(U)^2} \left| \frac{dK_{\mathrm n}[f(U)]}{dU} \right|^2 U_m^2 \operatorname{Re} \left( \frac{\dot K(u)(2\omega)\dot\beta(2\omega)} {1-\dot K[(U)(2\omega)]\dot\beta(2\omega)} \right) \cdot \operatorname{sign} \frac{dK[f(u)]}{du}, \tag{17} \]

or

\[ \frac{\Delta_1 K_{\mathrm n}(u)}{K_{\mathrm n}(u)} = 2k_{2\mathrm n}^2\operatorname{Re} \frac{\dot K(u)(2\omega)\dot\beta(2\omega)} {1-\dot K(u)(2\omega)\dot\beta(2\omega)} \operatorname{sign} \frac{dK[f(u)]}{du}; \tag{18} \]

\[ \varphi_{\mathrm{nz}} \approx \arcsin \frac{\Delta_2 K(u)}{K_{\mathrm n}(u)} = \arcsin \frac12\cdot \frac{1}{[K_{\mathrm n}(u)]^2} \left| \frac{dK_{\mathrm n}[f(u)]}{du} \right|^2 \cdot \]

\[ u_m^2\operatorname{Im} \frac{\dot K(u)(2\omega)\dot\beta(2\omega)} {K_{\mathrm n}(u)[1-\dot K(u)(2\omega)\dot\beta(2\omega)]}, \tag{19} \]

or

\[ \varphi_{\mathrm{nz}} \approx \arcsin 2k_{2\mathrm n}^2 \operatorname{Im} \frac{\dot K(u)(2\omega)\dot\beta(2\omega)} {K_{\mathrm n}(u)[1-\dot K(u)(2\omega)\dot\beta(2\omega)]} \operatorname{sign} \frac{dK[f(u)]}{du}. \tag{20} \]

Substituting into (5) the refined value of the transfer coefficient of the nonlinear element, we can determine the amplitude of the steady-state oscillations.

From the phase-balance condition we find the nonlinear frequency shift

\[ \Delta \omega \simeq \frac{-\varphi_{\mathrm e}} {\dfrac{\partial \varphi_{\mathrm e}(\omega)}{\partial \omega}} . \tag{21} \]

The correction to the magnitude of the 3rd harmonic of the generator is equal to

\[ \begin{aligned} \Delta \dot A_3 &= \frac{\dot K_{\mathrm p}(3\omega)} {1-\dot K(u)(3\omega)\dot\beta(3\omega)} \cdot \frac{1}{2\pi} \int_{0}^{2\pi} \left\{ (-K'_{\mathrm n})U_m \right. \\ &\qquad\left. {}\cdot \frac{ A_{2\mathrm n}\dot K(u)(2\omega)\dot\beta(2\omega) }{ K_{\mathrm n}(u)\,[1-\dot K(u)(2\omega)\dot\beta(2\omega)] } \cdot e^{j3\omega t} \right\} e^{-j3\omega t}\,d\omega t = \\ &= \frac{1}{2}\cdot \frac{1}{|K_{\mathrm n}(u)|^2} \left| \frac{dK_{\mathrm n}[f(u)]}{du} \right|^2 U_m^3 \frac{\dot K(u)(3\omega)} {1-\dot K(u)(3\omega)\dot\beta(3\omega)} \cdot \\ &\qquad{}\cdot \frac{\dot K(u)(2\omega)\dot\beta(2\omega)} {1-\dot K(u)(2\omega)\dot\beta(2\omega)} \operatorname{sign} \left\{ -\frac{dK_{\mathrm n}[f(u)]}{du} \right\} = \\ &= 2\kappa_{2\mathrm n}^{\,2}U_m \frac{\dot K(u)(3\omega)} {1-\dot K(u)(3\omega)\dot\beta(3\omega)} \cdot \frac{\dot K(u)(2\omega)\dot\beta(2\omega)} {1-\dot K(u)(2\omega)\dot\beta(2\omega)} \operatorname{sign} \\ &\qquad{}\cdot \left\{ -\frac{dK_{\mathrm n}[f(u)]}{du} \right\}. \end{aligned} \tag{22} \]

Comparison of the formulas obtained makes it possible, despite the simplifications admitted, to draw a number of important conclusions.

1. Finding a refined solution is justified only when

\[ \left| \kappa_{2\mathrm n}^{\,2} \frac{\dot K(u)(3\omega)/\dot K(u)(\omega)} {1-\dot K(u)(3\omega)\dot\beta(3\omega)} \cdot \frac{\dot K(u)(2\omega)\dot\beta(2\omega)} {1-\dot K(u)(2\omega)\dot\beta(2\omega)} \right| >2\kappa_{3\mathrm y}, \tag{23} \]

where \(K_{3\mathrm y}\) is the coefficient of the third harmonic of the amplifier (of the open-loop system).

If inequality (23) is replaced by equality, then even then the maximum value of the relative correction to the amplitude of the fundamental harmonic of the self-oscillations does not exceed 10%.

1. For stable generation the following conditions must be satisfied:

a) for an even nonlinearity

\[ \frac{dK[f(u)]}{du}\cdot \operatorname{Re} \frac{\dot K(u)(2\omega)\dot\beta(2\omega)} {1-\dot K(u)(\omega)\dot\beta(2\omega)} <0; \]

b) for an odd nonlinearity

\[ \frac{d^2K[f(u)]}{du^2}<0; \]

c) for a combined nonlinearity

\[ \frac{1}{2} \left\{ \frac{dK[f(u)]}{du} \right\}^{2} \cdot \operatorname{Re} \frac{\dot K(u)(2\omega)\dot\beta(2\omega)} {K_{\mathrm n}(u)[1-\dot K(u)(2\omega)\dot\beta(2\omega)]} \cdot \]

\[ \operatorname{sign} \frac{dK[f(u)]}{du} + \frac{3}{8}\cdot \frac{d^2K[f(u)]}{du^2} <0. \]

1. In order to obtain minimal distortion at a given stability of the oscillation amplitude, it is necessary that the NE have a symmetrical characteristic (odd).

1. For the given circuit, there is a one-to-one relationship between the sign of \(\dfrac{dK(U)}{dU}\) and the phases of the harmonics. This relationship also exists between the quantities \(K(u)\) themselves and \(K_{\mathrm{p}}\) \((K_{\mathrm{gu}})\).

References

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